Average Error: 8.0 → 1.7
Time: 4.2s
Precision: 64
\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \le -5.81880234558190669 \cdot 10^{134} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 1.95624520253703639 \cdot 10^{220}\right):\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right) - \left(t \cdot 4.5\right) \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} \cdot \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{2}\\ \end{array}\]
\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}
↓
\begin{array}{l}
\mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \le -5.81880234558190669 \cdot 10^{134} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 1.95624520253703639 \cdot 10^{220}\right):\\
\;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right) - \left(t \cdot 4.5\right) \cdot \frac{z}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{a} \cdot \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{2}\\

\end{array}
double code(double x, double y, double z, double t, double a) {
	return (((x * y) - ((z * 9.0) * t)) / (a * 2.0));
}
↓
double code(double x, double y, double z, double t, double a) {
	double VAR;
	if (((((x * y) - ((z * 9.0) * t)) <= -5.818802345581907e+134) || !(((x * y) - ((z * 9.0) * t)) <= 1.9562452025370364e+220))) {
		VAR = ((0.5 * (x * (y / a))) - ((t * 4.5) * (z / a)));
	} else {
		VAR = ((1.0 / a) * (((x * y) - ((z * 9.0) * t)) / 2.0));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original8.0
Target5.6
Herbie1.7
\[\begin{array}{l} \mathbf{if}\;a \lt -2.090464557976709 \cdot 10^{86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a \lt 2.14403070783397609 \cdot 10^{99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (- (* x y) (* (* z 9.0) t)) < -5.818802345581907e+134 or 1.9562452025370364e+220 < (- (* x y) (* (* z 9.0) t))

    1. Initial program 24.6

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]

    2. Taylor expanded around 0 24.2

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t \cdot z}{a}}\]
    3. Using strategy rm

    4. Applied *-un-lft-identity24.2

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t \cdot z}{\color{blue}{1 \cdot a}}\]

    5. Applied times-frac14.1

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \color{blue}{\left(\frac{t}{1} \cdot \frac{z}{a}\right)}\]

    6. Applied associate-*r*14.2

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{a} - \color{blue}{\left(4.5 \cdot \frac{t}{1}\right) \cdot \frac{z}{a}}\]

    7. Simplified14.2

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{a} - \color{blue}{\left(t \cdot 4.5\right)} \cdot \frac{z}{a}\]
    8. Using strategy rm

    9. Applied *-un-lft-identity14.2

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{\color{blue}{1 \cdot a}} - \left(t \cdot 4.5\right) \cdot \frac{z}{a}\]

    10. Applied times-frac2.9

      \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{x}{1} \cdot \frac{y}{a}\right)} - \left(t \cdot 4.5\right) \cdot \frac{z}{a}\]

    11. Simplified2.9

      \[\leadsto 0.5 \cdot \left(\color{blue}{x} \cdot \frac{y}{a}\right) - \left(t \cdot 4.5\right) \cdot \frac{z}{a}\]

    if -5.818802345581907e+134 < (- (* x y) (* (* z 9.0) t)) < 1.9562452025370364e+220

    1. Initial program 1.1

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity1.1

      \[\leadsto \frac{\color{blue}{1 \cdot \left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)}}{a \cdot 2}\]

    4. Applied times-frac1.2

      \[\leadsto \color{blue}{\frac{1}{a} \cdot \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{2}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \le -5.81880234558190669 \cdot 10^{134} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 1.95624520253703639 \cdot 10^{220}\right):\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right) - \left(t \cdot 4.5\right) \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} \cdot \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, I"
  :precision binary64

  :herbie-target
  (if (< a -2.090464557976709e+86) (- (* 0.5 (/ (* y x) a)) (* 4.5 (/ t (/ a z)))) (if (< a 2.144030707833976e+99) (/ (- (* x y) (* z (* 9 t))) (* a 2)) (- (* (/ y a) (* x 0.5)) (* (/ t a) (* z 4.5)))))

  (/ (- (* x y) (* (* z 9) t)) (* a 2)))